LA IMPORTANCIA DEL IMPUESTO DE TENENCIA Y USO DE VEHÍCULOS EN EL D.F Y EN QUERÉTARO

is Wrong

In fact the discussion was started in the previous Section 4. But let us go back to the historical roots that should help better under- standing of the facts that have been gradually established since K41 theory release. It was soon noticed that the reasoning leading to the K41 spectrum is not unique. Indeed, from dimensional considera- tions one can use< >2/3for the derivation of the spectral function (2.20). But equally it can be local(r, t) and not the averaged< >

used from dimensional considerations. But(r, t) is itself a fluctuat- ing field. What it means is that dimensionally we could use for the spectrum definition, say < n _>2/3n_{, where n is arbitrary.}31 _There

are no physical or mathematical reasons to choose n = 1. If was similar to a normal statistical type quantity fluctuating as a Gaus- sian variable this would be not a significant change affecting only a constant in front of the spectrum. However it was early noticed by Batchelor and Townsend (1949) that this is not the case. In fact is quite a wildly fluctuating quantity that reminds nothing of the usual field fluctuations in near to equilibrium statistical systems. In Fig. 7borrowed from Kit, et al., (1987) one can see typical experimental

31_{This reasoning is usually attributed to L. Landau and can be found in B.}

time traces of various quantities measured at a location in turbulent flow past a grid-a velocity component denoted as ui(t) , a vorticity

componentωi, a part of the helicity densityui(t)ωi(t), energy density

dissipation(t), enstrophyω2_{(t) and other more nonlinear quantities}

relevant for turbulent dynamics. Inspecting the time traces immedi- ately shows that for instance the velocity component varies in time in what appears a relatively smooth Gaussian like manner. But the en- ergy dissipation and enstrophy time traces are totally different. They consist of quiescent long signals with quite rare distinct large ampli- tude peaks at approximately the same locations on the time axis for the two fields. Even more distinct peaks are evident for other quanti- ties depicted in theFig. 4that are cubic nonlinearities with respect to the velocity field derivatives.

It is clear that dependent onnthe averages< n_>2/3n _{would be}

completely different and in consequence the energy spectrum may be different. Since there is no reason to choose onenor another for the derivation of the energy spectrum the whole sequence of considera- tions that led to K41 theory crumbles.

The velocity fluctuations in turbulent flows are primarily deter- mined by large scale or low wavenumbers velocity harmonics as is seen from Eq. (2.24). In contrast the fluctuations of the velocity derivatives in general and the enstrophy, its rate of growth and en- ergy dissipation rate < (t) are all strongly skewed to small scales or high wavenumbers values, see for instance Eqs. (2.23), (3.28) and (3.29). Note that the last property would remain the same if the en- ergy spectrum is not K41 but another power law close enough to K41. The moments of the energy dissipation rate< n>2/3nare definitely anomalous by comparison with what is regarded as normal in statis- tical analysis that is the moments as they should have been if their fluctuations followed the normal Gaussian (bell curve) distribution. In fact for big enough they are defined primarily by the rare peaks of intensity in the time traces and this dominance increases with the growth of n. But at the same time this anomalous growth can be only due to the high wave numbers harmonics of (t). Together it means that the high wave numbers harmonics of the energy dissipa- tion(t) are located in the time trace amplitude peaks or equivalently in narrow bands of physical space flow volume.32 _{The same is true}

for all other velocity derivatives related quantities such as enstrophy, enstrophy generation, etc., as can be clearly seen inFig. 7 below.

So far it was the usual description of intermittency familiar to everyone stuyding turbulence and chaos in dynamical systems. But the reality is that these intermittent patches of activity are built of coherent helical cells with near to maximal helicity and besides forming obvious clusters of organized vortical activity. The coherent objects that I namedBCCabove. The recognition ofBCCindicates a totally new reality of understanding intermittency in turbulence.

Let us try to develop this understanding step by step. To start with after years of experimental study it is rather safe to believe that itself fluctuates in accordance with a scaling power law as follows, e.g., Monin and Yaglom (1975):

<((0)−< >)2>=< >(L/r)µ_{, (5.1)}

whereLis the integral scale and the experimental value of parameter

µ are not really precise, but probably in the interval 0.3 < µ <

0.5. In particular forr=ld one obtains from (5.1) the mean square

deviation of(t) that is actually measured in experiments:

<((0)−< >)2>=< >(L/ld)µ=< >Re3µ/4, (5.2)

Other quantities often measured in the laboratories are the so-called structure functions of longitudinal velocity projections.33 _{These are}

defined as follows:

<∆v(r)n_l >=<[v(r)−v(0)]_ln>∝< >n/3(r/L)n/3−µν(n)_{, (5.3)}

where the subscript l means longitudinal projection. The meaning of the relation (5.3) is like this. If K41 is correct in its original form then µ = 0. This is because the original K41 theory ignores the

anomalously large fluctuations of small scale velocity variance and the related quantities, such as space and time derivatives of v(r, t) and their powers. In other words the original K41 ignores intermittency. This would mean that <v(r)n

l >=< >n/3 rn/3. But experiment a rudimentary ergodicity hypothesis.

33_{Only the longitudinal projections of the velocity variations parallel to the hot}

wire are usually measured in experiment. The only exceptions that I am aware are the experimental works sited above (see also Endnoteg).

Figure 7: Laboratory measurements of time series of various turbu- lent quantities in turbulence past the grid from Kit, et al., (1987). From top to bottom they are one velocity component u1, one vor-

ticity component ω1, u1ω1, energy dissipation, rate , enstrophy ω2,

part of the vorticity stretching term,sij = 2eij, see Eq. (1.18), etc.

We chose these particular measurements because they were done in a unique set of experiments in turbulent electrolyte. This allowed direct measurements of the quantities in a manner different to the usual hot wire anemometer measurements in turbulence that neces- sitate further application of various assumptions in order to arrive at the quantities of physical interest. It can be clearly seen that the fluctuations of velocity are mild, near to Gaussian, vorticity is much more intermittent with distinct peaks separated by long stretches of mild fluctuations, which is seen even better from observing fluctua- tions of enstrophy. The higher is the power of velocity derivatives the more obvious are the rare peaks of intensity separated by quiescent periods and this is the characteristics of intermittency. The higher are the orders of moments the more dominant will be the contribution of the rare peaks of intensity of the respective quantities.

firmly shows that although µν(n) ≈0 for n = 2 with experimental

0 and is nonlinear at least for 2 < n ≤ 6.34 _{Intermittency thus}

conforms with generalized scaling.

This latter is a relaxed version of the K41 theory that stated that the properties of turbulence in the inertial range should be indepen- dent of both the Reynolds number, in the limit of Re→ ∞and the integral scale L. If the first and more profound scaling assumption is still in place but the second is relaxed there is freedom now to use the L parameter for the scaling power laws of the type of (5.1) and (5.3). In other words the velocity structure functions become more and more singular for small separation scales r → 0, i.e., or in general sense for high wavenumbers. This is in accord with the vision that the intense part of turbulence connected with turbulent field velocity variations and derivatives is concentrated in progres- sively smaller and smaller sub-domains in the fluid volume.35 Thus it seems that although the original K41 theory is not really correct nevertheless the generalized scaling concept and universality of tur- bulence, at least HIT, are confirmed experimentally with reasonable experimental evidence.

The above led to great efforts of building phenomenological mod- els that would put together K41 theory and intermittency. The gen- eralized scaling theories inevitably lead to fractal and multifractal models of turbulence (Mandelbrot1977, 1982, 1983). Let us consider for methodological purposes the simplest of them due to Novikov and Stewart (1964), which is known in literature as fractally homogeneous turbulence-FHT. While doing this we will introduce some important definitions that will be used for the exposition of a dynamical theory later in this paper.

Consider a box of size L that is split into sub-boxes of size l1.

If we measure the energy flux (r, t) , or better to say observe it with low resolution vision, for instance our sight is clouded by tears and we see no fine details we may conclude that the dissipation is

34_{Genrallynis not integer. The value ofn= 6 is approximate and seems not}

universal. The behaviour ofµν(n) for highnmay be linear but this is likely due

to the fact that high order moments are spurious and do not have much physical meaning if at all as discussed below.

35_{Note thatµin (5.1) generally does not coincide with any ofµ}

ν(n) from (5.3),

unless additional assumptions are made. The intermittency exponents for the correlation functions in Fourier space are difficult to relate to those in physical space because the corresponding physical space Fourier transforms are not local.

homogeneously distributed everywhere in the whole box of the size

L. More precisely we would see the averaged over the cube and time value of the fluxV−1_T−1R

(r, t)dV dt, whereV is the full size cube volume andT =T(V) is the corresponding time scale. But when we improve our resolution, wipe mist from our eyes, we would see that in fact the dissipation occurs, may be also homogeneously, but only

inn < mof the sub-boxes of sizel1. In other words we would observe

now the flux averaged over only a part of the space/time in a fraction

n/m= (l/L)µ <1 sub-boxes. If we magnify our resolution further, put on the glasses, we would see that in reality the dissipation occurs in even a smaller fraction of sizel2 such that the ratio λ=l2/l1 =

l1/L=const. The fractally homogeneous set is built by self-similar

iterations of this process. Taking account of the constancy of the total energy flux through the space sub-boxes with increasingly finer resolution the following relation follows:

(L) =(l1)n/m=(l2)(n/m)2=...=(li)(n/m)li, (5.4)

or:

i=i−1λ−µ=0(l1/L)−µ. (5.5)

The effective volume through which the energy flux is passing (and eventually becomes equal to the viscous rate of energy dissipation at the scales of orderld) is determined in a similar way with the result:

Vi=L3(n/m)i=L3(l1/L)µ(l2/l1)µ...(li/li−1)µ=L3(l1/L)µ.

(5.6) Now it is easy to calculate the moments of the energy flux:

<((0)−< >)((r)−< >)n>=< >n(L/l1)(n−1)µ;r≈1.

(5.7) In particular for n = 2 setting l1 = ld we obtain the scaling rela-

tion (5.2). We built a simplest possible fractal model with the ”vol- ume” tending to zero with l1/L→ld/L→0 asRe−3µ/4 →0, while

the volume of the active sub-domain tends to zero asL3_Re3µ/4_{, the}

area of the surface bounding this sub-domain will tend to infinity as L2_Re3µ/4

. The actual dimension of the sub-domain is a fractal

DF =D−µwithD= 3 in this case.

Let us consider the above a bit differently in a way convenient for the further exposition in the sections below. Let us associate a

material volume with an active sub-domain embedded in 3Dflow do- main. Let us make an infinitesimal scale transformation r0 =e−l_r,

where we choose l → 0 and positive. By definition of a fractal set of dimension DF =D−µembedded in 3D space, the material vol-

ume of a small sub-domain under this scale transformation becomes ∆DF_{r= e}−(D−µ)l_∆D_{r. If the fractal is isotropic it means that the}

effective contraction under the scale transformation is the same in

x, y, z directions, i.e., r → e−µl/3r0 = re−(D−µ)l/3. Let us iterate the rescaling m → ∞ times. Since m is arbitrary let us choose it in such a way that limm→∞,l→0e−ml = ln(r/ld). Then asymptot-

ically the iteration of infinitesimal scaling transformation results in the following:

∆Dr→∆DF_{r= ∆}D−µ_rlµ

d, (5.8)

while

r→r(D−µ)/3l_dµ/3.

But it can be that the fractal is strongly anisotropic. In the extreme case the scaling will look like this:m

∆Dr→∆DF_{r= ∆r}(D−µ)_lµ

d,

(x, y)→(x, y)0, z→z(D−µ)lµ_d. (5.9) Let us associate a physical field with the material volume, for instance the energy flux. Since the latter is conserved under the scaling trans- formation (5.8) we obviously arrive at (5.7). But what happens with the velocity field v(r, t) and other fields that are not conserved un- der the scale transformation (5.8) or (5.9)? In the FHT it is easy to calculate. One of the few exact consequences of the Navier-Stokes equations is a fairly remarkable relation of Karman-Howarth-Dryden for the velocity structure function of order n = 3 (see also (2.10)). If statistical homogeneity and isotropy are assumed then it can be derived for the separation distancesr from the inertial range and in the limitRe→ ∞(e.g., Monin and Yaglom, 1975):

<∆v3>= 4/5< > r. (5.10)

Since the cascade remains homogeneous at each cascade subdivision in (5.4) it is natural to suppose that the same relation is preserved at each scale. Then it is easy to derive that the exponents in (5.3)

are linear and all determined by one exponentµdefining the square fluctuations of (r,):

µν(n) =µ(n−3)/3, (5.11)

implying corrections to the second order structure function atn= 2 and subsequently the K41 spectrum. FHT is the simplest fractal model of turbulence that can only be by the criteria of our times. More complicated phenomenological so-called lognormal model was built by Kolmogorov and Obukhov. In the lognormal model (e.g., Monin and Yaglom, 1975) the intermittency parameterµ(n) is non- linear. Realization that the lognormal model of turbulence is a par- ticular case of more general class of multifractal models of turbu- lence came much later (e.g., Parisi, Frisch, 1985). For quantitative and deep analysis one should go to Mandelbrot monograph and the works of many authors cited therein. For the purposes of this paper the most relevant analysis of multifractals, their conceptual founda- tions and relevance for turbulence, geophysics and meteorology have been developed in the seminal works of Lovejoy, Schertzer and their co-authors whom I cite often in this paper (e.g., Lovejoy, 1982; Love- joy and Schertzer, 1985; Lovejoy, et., al., 2007; Lovejoy, et., al., 2008; Lilley, et., al., 2008; Schertzer and Lovejoy, 1983, Schertzer and Love- joy, 1985a; Schertzer and Lovejoy, 1985b, etc.). Most of their works can be conveniently found on the website of GANG-Group for the Analysis of Nonlinear Variability in Geophysics.

It seems beyond reasonable doubt that based on the analysis of extensive geophysical data, airborne, satellite, radar and recently li- dar that all the atmospheric turbulent fields of importance, turbulent wind in horizontal and vertical, rainfall, admixtures distribution in at- mosphere, etc., have universal scaling multifractal organization. The more extreme are the deviations of these fields from the mean the smaller is the dimensional support for these extremals. This scaling organization extends from the smallest scale of millimeters to appar- ently the largest planetary scales spanning ten orders of magnitude, a truly remarkable conclusion contradicting all the previous geophys- ical concepts. At the same time it seems that there is no end to the extreme deviations of turbulent fields from their mean values. Most probably the Pdf for the extreme fluctuations is a power law; to be sure asymptotically in the limit of Re→ ∞. What it means is that

that the high order statistical moments of the fluctuating turbulent fields are divergent and actually do not have physical meaning in this limit. Since experiment is always done for finite values of the measurements would show finite values for all orders of statistical moments. But these finite values are spurious and do not have sense beyond the fact that the moments are dominated by a few extreme deviations from the mean. This is good news for practitioners of turbulence. Because if multifractal structure with all meaningful mo- ments of the velocity field and its derivatives is correct then generally turbulence must be characterized by an infinity of scaling exponents for different physical fields and their different order moments. This would render the whole concept quite useless and would forever make turbulence intractable to any reasonable scientific analysis. In fact the realization that only a finite number of moments of physical fields in turbulent flows have meaning makes it possible trying to determine their dynamical significance and building a dynamical theory.

In the framework of helical structures concept the statistical de- scription of turbulence is limited and it seems clear that high order statistics should be meaningless. Indeed, what is the meaning of the statistical moments if the main contribution to these moments comes fromBCC, an immensely coherent and asymptotically fractal object? So far the singularities of high order moments of turbulent fields have not been seen in laboratory experiments, e.g., Sreenivasan and Antonia (1991), but this does not at all eliminate the concept’s validity since the Reynolds numbers of laboratory measurements, the precise ones, are incomparably lower than in geophysical flows. On the other hand it is increasingly indicative from DNS that reaching the real scaling is a slow asymptotic process requiring high values of Re. Also, the local in space nature of laboratory measurements are not conducive for capturing the extreme fluctuations that may occupy a very small physical sub-domain in space/time.n

Let us come back to the role of helicity. The HIT and statisti- cal mirror symmetry are assumed below, meaning that < h >=<